Tensor codes have been extensively studied in theoretical computer science because they provide natural constructions of locally testable codes [1]}, [2]}, [3]}, [4]}. A code \(\mathcal {C}\)  is locally testable if it has a tester algorithm that makes a small number of queries to a given \(w: [n] \rightarrow \Sigma \) , accepts if \(w\)  is a codeword of \(\mathcal {C}\) , and rejects with noticeable probability if \(w\)  is far from every codeword. In other words the code has efficient probabilistic tests for membership. Locally testable codes are central components of probabilistically checkable proofs (PCPs); in particular improvements in PCPs have been continually driven by the the development of improved low-degree tests, which are tests for codes based on low-degree polynomials [5]}, [6]}, [7]}, [8]}.
